Intuitionism and Intuitionistic Logic | Research & Encyclopedia Articles

Intuitionism and Intuitionistic Logic

Logic, in the modern preponderantly mathematical sense, deals with concepts like truth and consequence. The main task of logic is to discover the properties of these concepts. Ever since Aristotle it had been assumed that there is one ultimate logic for the case of descriptive statements, which lent logic a sort of immutable, eternal appearance. Only in the beginning of the twentieth century were certain principles of traditional logic submitted to a critical revision. It was L. E. J. Brouwer, who, in a radical constructive framework of mathematics, discovered that traditional logic could not be upheld in its full extent.

This entry sketches the basic ideas of Brouwer's constructivism, which goes by the name of intuitionism, and then discusses the fundamental principles. Next, an exposition of the familiar notions, such as proof system, semantics, and the like, is provided. In particular, this entry will show how the Brouwerian mathematical universe takes a special place in terms of its logical properties.

Intuitionistic Truth

For all practical purposes it suffices to consider in mathematical logic only a few logical constants, or connectives. The traditional conjunction (and, ∧), disjunction (or, ∨, and not, ¬), and implication (if …, then … , →) will do for propositional logic.

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